#27382202

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#108×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #20459

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#20459
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(61,74)(66,80)(71,85)(75,76)(78,88)(81,82)(84,89)(86,87);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,55)(58,61,64,74)(59,66,69,80)(60,71,73,85)(62,75,63,76)(65,78,79,88)(67,81,68,82)(70,84,83,89)(72,86,77,87);(58,60,63,72,64,73,62,77)(59,65,68,70,69,79,67,83)(61,71,76,86,74,85,75,87)(66,78,82,84,80,88,81,89);(54,55,57,56)(58,59,62,67,64,69,63,68)(60,70,72,79,73,83,77,65)(61,66,75,81,74,80,76,82)(71,84,86,88,85,89,87,78)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,e^b=eh,f^b=fh,d^c=dh,i^d=i^52

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 167678);